View source for Proportional Subset Voting

Proportional Subset Voting (PSV) is a [[Single Member system|Single-Winner]] and [[Multi-Member System|Multi-Winner]], [[Cardinal voting systems]] proposed by [[User:Aldo Tragni|Aldo Tragni]].

==Procedure==

Ballot uses range [0%,100%]. <math>N</math> is the number of winners.

For each vote, and for each subset of <math>N</math> candidates:
* create a list of <math>G_i</math> groups, 1 for each percentage, multiple of <math>\frac{100\%}{N}</math> ''(eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%)''.

* for each <math>G_i</math>, the voters with a total approval <math>=G_i</math> are counted getting <math>C_i</math> <math>(\ \mathrm{ total\ approval\ of\ the\ N\ candidates}=\frac{\mathrm{ratings\ sum}}{N}\ )</math>. Each <math>C_i</math> is then divided by the number of voters.

* find the group with the lowest percentage <math>G_{min}</math> containing <math>C_{min}>0</math>, and associate <math>S_{min}=\{G_{min},C_{min}\}</math> to the subset.

* find the group with the highest percentage <math>G_{max}</math> containing <math>C_{max}>0</math>, and associate <math>S_{max}=\{G_{max},C_{max}\}</math> to the subset.

Sort the subsets from major to minor based on the <math>S_{min}</math> value; if there are tied subsets then sort them from minor to major based on the <math>S_{max}</math> value. The subset that is first after sorting wins.

If there are any tied subsets, <math>P</math> is calculated which is the sum of the products between <math>G_i</math> and <math>C_i</math>. The subset that has highest <math>P</math> wins.

===Procedure with range===

The ratings of the range have values between [0%,100%]. For example, if it has 5 ratings {0,1,2,3,4} then the respective values will be {0%,25%,50%,75%,100%}.

The only difference with the procedure that uses multiple-choice ballots is that:

* the total approval of a voter for N candidates must be somehow rounded up to a multiple of <math>\frac{100\%}{N}</math>, in order to be counted in the list of <math>G_i</math> groups.

==Example==

Ballot uses 2 ratings, that is: {0,1} = {0%, 100%}. There are 6 winners and the following votes:

 A[1] B[1] C[1] D[1] E[1] F[1] G[0] ...
 A[1] B[1] C[1] D[1] E[1] F[1] G[0] ...
 A[1] B[1] C[1] D[0] E[0] F[0] G[1] ...

The list of groups associated with the subset of 6 candidates is:

 0%[] 16%[] 33%[] 50%[] 66%[] 83%[] 100%[]

The total approval of the 3 voters is calculated for the following subsets:

 ABCDEF: [100%, 100%, 50%]
 ABCDEG: [83%,  83%,  66%]

Total approvals are counted for each subset, and then divided by the number of voters (3 in this case):

 ABCDEF
  Count:    0%[] 16%[] 33%[] 50%['''1''']   66%[] 83%[] 100%['''2''']
  Division: 0%[] 16%[] 33%[] 50%['''33%'''] 66%[] 83%[] 100%['''66%''']
  Short:    50%['''33%'''] 100%['''66%''']
 ABCDEG
  Count:    0%[] 16%[] 33%[] 50%[] 66%['''1''']   83%['''2''']   100%[]
  Division: 0%[] 16%[] 33%[] 50%[] 66%['''33%'''] 83%['''66%'''] 100%[]
  Short:    66%['''33%'''] 83%['''66%''']

Find <math>\{S_{min},S_{max}\}</math> and sort:

 ABCDEG: {33%,83%}
 ABCDEF: {33%,100%}

ABCDEG wins.

Calculate P, just to show how this is done:

 ABCDEF: 50%[33%] 100%[66%]  -->  50%*33% + 100%*66% = 0,825
 ABCDEG: 66%[33%] 83%[66%]   -->  66%*33% + 83%*66%  = 0,765

==Subset Voting (category)==

N is the number of winners.

For each vote, and for each subset of N candidates, a score S is obtained using procedure p1, finally obtaining the converted votes.

Procedure p2 (eg a [[Single Member system|Single-Winner system]]) is used, on the converted votes, to obtain the winning subset.


''The size of the range, procedure p1, and procedure p2 chosen, determine the variant of Subset Voting.''

[[Category:Single-winner voting methods]]
[[Category:Multi-winner voting methods]]
[[Category:Cardinal voting methods]]